AbstractWe investigate the maximal open domain $${\mathscr {E}}(M)$$
E
(
M
)
on which the orthogonal projection map p onto a subset $$M\subseteq {{\mathbb {R}}}^d$$
M
⊆
R
d
can be defined and study essential properties of p. We prove that if M is a $$C^1$$
C
1
submanifold of $${{\mathbb {R}}}^d$$
R
d
satisfying a Lipschitz condition on the tangent spaces, then $${\mathscr {E}}(M)$$
E
(
M
)
can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is $$C^2$$
C
2
or if the topological skeleton of $$M^c$$
M
c
is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a $$C^k$$
C
k
-submanifold M with $$k\ge 2$$
k
≥
2
, the projection map is $$C^{k-1}$$
C
k
-
1
on $${\mathscr {E}}(M)$$
E
(
M
)
, and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion $$M\subseteq {\mathscr {E}}(M)$$
M
⊆
E
(
M
)
is that M is a $$C^1$$
C
1
submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with $$M\subseteq {\mathscr {E}}(M)$$
M
⊆
E
(
M
)
, then M must be $$C^1$$
C
1
and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between $${\mathscr {E}}(M)$$
E
(
M
)
and the topological skeleton of $$M^c$$
M
c
.
Existence, uniqueness and regularity of the projection onto differentiable manifolds
